An efficient numerical method on modified space-time sparse grid for time-fractional diffusion equation with...

Area of Science:

• Numerical analysis
• Partial differential equations
• Computational mathematics

Background:

• Time-fractional diffusion equations (TFDEs) are crucial in modeling anomalous diffusion processes.
• Low regularity of TFDE solutions, often due to non-smooth initial conditions, significantly hinders numerical method convergence.
• Existing numerical methods struggle to maintain accuracy and efficiency when dealing with the low regularity inherent in TFDE solutions.

Purpose of the Study:

• To develop a highly efficient algorithm for solving d-dimension time-fractional diffusion equations (TFDEs).
• To enhance the convergence rate of numerical methods for TFDEs, particularly addressing challenges posed by low-regularity solutions.
• To introduce and analyze a modified space-time sparse grid (STSG) method tailored for TFDEs.

Main Methods:

• Development of a space-time sparse grid (STSG) method using sine basis for spatial discretization and linear element basis for temporal discretization.
• Construction of the STSG through a tensor product of spatial multilevel and temporal hierarchical bases.
• Integration of a full grid approach into the STSG to create a modified STSG, addressing accuracy issues with rapidly changing initial solutions.

Main Results:

• The standard STSG method can achieve a specific accuracy order with a reduced number of degrees of freedom (DOF) under certain conditions.
• The modified STSG method demonstrates improved accuracy and robustness, particularly for TFDEs with solutions that change rapidly at the initial moment.
• Comparative numerical experiments validate the significant advantages of the modified STSG method over standard approaches.

Conclusions:

• The modified STSG method provides a robust and efficient numerical scheme for solving d-dimension TFDEs.
• This approach effectively overcomes the limitations of standard STSG methods when dealing with low-regularity solutions.
• The developed algorithm offers a promising tool for the accurate simulation of anomalous diffusion phenomena.